Chapter 6 Inductive limits

Proposition 6.2.4
Every inductive sequence of -algebras has an inductive limit , in addition, the following all hold.

  1. (Where the index is {to,from})
  2. If and are as in the uniqueness property of inductive limit, then;
    (a) for all .
    (b) is injective if and only if
    (c) is surjective if and only if

There is a similar result for inductive limits of abelian groups, this can be applied to the compatibility of the functor with inductive limits.


Proposition 6.2.5.
Each sequence of abelian groups has an inductive limit moreover, the following hold.

  1. for each .
  2. Same as in Proposition 6.2.4. No closure needed though.

Lemma 6.3.1
Let be a -algebra,

  1. If is self-adjoint, such that , then there exists a projection such that .
  2. Let . If there exists an element with and then . In the murray von Neumann sense.

Theorem 6.3.2 (Continuity of K0)
For each inductive sequence of -algebras, and are isomorphic as abelian groups.


Definition 6.4.a (Stabilization)
The stabilization of -algebras. We define the stabilization of to be , where denotes the compact operators of for .
Instead of working with tensor products, it is more useful and insightful to work with a different realization of where we build the sequence Where the connecting maps are are given by for each . Let be the limit of its sequence.


Proposition 6.4.1 (Stability of K0)
Let be the canonical inclusion of a -algebra into its stabilization . Then is an isomorphism.